Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain

Abstract

Finding a target in a complex environment is a fundamental challenge across natural systems, from chemical reactions to sperm cells reaching an egg. A powerful strategy to reduce search times is redundancy: deploying many independent searchers increases the probability that at least one succeeds, particularly when success is driven by rare events. When the underlying stochastic motion features broadly distributed step lengths, rare long relocations dominate the dynamics, making redundancy especially effective. Here, we investigate the statistics of extreme events for the mean first passage time in a system of $N$ independent walkers performing power-law distributed jumps with finite velocity, where target-reaching events are governed by single large fluctuations. We show that the mean first passage time of the fastest walker scales as $⟨T_N⟩\sim 1/N$, representing a dramatic speed-up compared to classical Brownian motion, and saturates at the minimum value $X/v$. We further extend the model to include random velocity. For fixed $N$, we identify a crossover, governed by a critical tail exponent ${\alpha }_c$, separating a regime dominated by a single large fluctuation (big jump) from a regime characterised by Gaussian extreme-value statistics arising from finite sampling effects. From these results, we derive a scaling law that links the number of walkers $N$ to the size $X$ of the search region. Our results demonstrate how redundancy, combined with rare-event statistics, can efficiently organise target-search processes in complex biological environments. As a prototypical example, we consider mammalian fertilization and derive, within a coarse-grained description, a cross-species scaling relation between the number of spermatozoa and the typical uterine size.

16 pages, 6 figures

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